The boundary of negatively curved groups

Gromov's article [Gr] contains fundamental properties of negatively curved groups. Several sets of seminar notes are available [FrN, SwN, USN] that contain more detailed accounts of, and further expand on, Gromov's article. A sequence Xi E r, i = 1, 2, ... , is convergent at infinity provided (Xi' x) -t 00 as i, j -t 00. Two sequences {Xi}' {YJ convergent at infinityare equivalent if (Xi' Yi) -t 00 as i -t 00. A point in the boundary ar of r is an equivalence class of sequences convergent at infinity. Now r u ar has a natural topology, in which r is a discrete subspace, and a typical (not necessarily open) neighborhood of a point a E ar represented by a sequence {xJ is given by {y E r u arj(a . y) > N} ,where (a· y) = lim(xi . y) if y E r, and (a· y) = lim(xi . Yi) if YEar is represented by a sequence {yJ. ar is a compact, metrizable, finite-dimensional space [Gr, SwN]. The important tool that relates ar with the cohomological properties of r is the Rips complex Pd(r). For every d ~ 0, Pd(r) is the simplicial complex whose vertices are elements of r, and a collection XI' ••• , xk E r spans a simplex if d(Xi' x) ~ d for all i, j. The natural group action of r on itself by left translations gives rise to an action on Pd(r). The key observation of Rips is that when d is sufficiently large, Pd(r) is contractible and therefore, when r is torsion free, provides a model for Er. More precisely, we have Proposition 1.1 [Gr, Lemma 1.7 .A; SwN, §4.2, Proposition 9]. Suppose that r is a negatively curved group, and let t5 be as in the definition. Choose an integer